A generalization of the notion of Riemannian submersion is given for submersions of a Lorentzian manifold onto a Riemannian manifold of one lower dimension. These maps are called comersions. A comersion induces a surjection of the space of causal curves in the domain manifold onto curves of bounded length in the image manifold. Thus if the source space is causally 1-connected, the target space is 1-connected.

1. We suppose that f(t) is integrable L and periodic with period 2π and we denote the Fourier series of f(t) at the point t = x by

We denote the nth partial sum of the Fourier series by sn. We will prove here a general result (i.e Theorem 1) and some interesting results in connexion with associated series of a Fourier series of the type

We write

The object of the present paper is to prove the following theorems. (In all the following cases k is any constant > π if nothing else is mentioned.)

Modifications to the Feynman propagation functions Sp and Dp are suggested in order to deal with field theoretic problems involving the presence in the initial and the final states, of free fermions or bosons, having all values of momentum (and energy) within a certain specified range.

Let K be a number field or a function field of characteristic 0, let φ ∈ K(z) with deg(φ) ⩾ 2, and let α ∈ ℙ1(K). Let S be a finite set of places of K containing all the archimedean ones and the primes where φ has bad reduction. After excluding all the natural counterexamples, we define a subset A(φ, α) of ℤ⩾0 × ℤ>0 and show that for all but finitely many (m, n) ∈ A(φ, α) there is a prime 𝔭 ∉ S such that ord𝔭(φm+n(α)−φm(α)) = 1 and α has portrait (m, n) under the action of φ modulo 𝔭. This latter condition implies ord𝔭(φu+v(α)−φu(α)) ⩽ 0 for (u, v) ∈ ℤ⩾0 × ℤ>0 satisfying u < m or v < n. Our proof assumes a conjecture of Vojta for ℙ1 × ℙ1 in the number field case and is unconditional in the function field case thanks to a deep theorem of Yamanoi. This paper extends earlier work of Ingram–Silverman, Faber–Granville and the authors.

Let G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_p$[G]-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser–Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of the representation ring for which regulator constants are defined. For all G, we show that this pairing is never identically zero. For formal reasons, this pairing will, in general, have non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to considering permutation lattices, then we show that the pairing is non-degenerate modulo the formal kernel. Using this we can show that, for certain groups, including dihedral groups of order 2p for p odd, the isomorphism class of any $\mathbb{Z}_p$[G]-lattice whose extension of scalars to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$, and a cohomological invariant of Yakovlev.

Envelopes of hypersurfaces arise in many different geometric situations. For example, the canal surfaces associated with a space curve can be obtained as the envelope of spheres of a fixed radius centred on that curve. (Other examples can be found in [11], chapter II and [4], chapter 5). In what follows we shall be entirely concerned with the local structure of such envelopes. So essentially we have a smooth map germ with 0 a regular value of the restriction of F to . For near 0 we have a smooth hypersurface near , where . The envelope of this family of hypersurfaces is obtained by eliminating t from the equations . Moreover, it can be identified with the discriminant of F viewed as an unfolding, by the x variables, of the function germ . This observation, together with standard results on versal unfoldings of functions, allows one to determine the local structure of many ‘generic’ geometric envelopes. See [2] for details.

In (2), Hall considered the question: for what varieties of soluble groups do all finitely generated groups satisfy max-n (the maximal condition for normal subgroups)? He has shown that the variety M of metabelian groups and more generally the variety of Abelian-by-nilpotent-of-class-c (c ≥ 1) groups has this property; whereas on the contrary, there are finitely generated groups in the variety V of centre-by-metabelian groups (i.e. defined by the law [x, y; u, v; z]) which do not satisfy max-n. One naturally raises the question: for what subvarieties of V do all finitely generated groups satisfy max-n?

In their 1990 paper, Bloch and Kato described the image of the Kummer map on an abelian variety over a local field, as the group of 1-cocycles which trivialise after tensoring by Fontaine's mysterious ring BdR. We prove the analogue of this statement for the universal nearly-ordinary Galois representation. The proof uses a generalisation of the Tate local pairing to representations over affinoid K-algebras.

We extend to a wider class of rearrangement invariant spaces a result of S. Guerre concerning the extraction of almost symmetric sequences from a given weakly null sequence in Lp.

The two fourth-order partial differential equations giving the transverse displacement and the stress functions for shallow shells are reduced to a sixth-order differential equation in one variable which has been solved, and the values of displacements, stress resultants and couples are all expressed in terms of Bessel Functions of imaginary argument. Numerical values of the displacement presented for points on the shell on different meridian lines, showed that on the symmetrical line deflexion is maximum at the point of loading but the point of maximum deflexion shifts gradually towards the pole.

Let q be a prime power and let G be a group acting faithfully and vertex transitively on a graph such that for each vertex x, the stabilizer Gx is finite and contains a normal subgroup inducing on the set of neighbours of x a permutation group isomorphic to the linear group L5(q) acting on the 2-dimensional subspaces of a 5-dimensional vector space over Fq. In a companion paper, it is shown, except in some special situations where q = 2, that the kernel of the action of a vertex stabilizer Gx on the ball of radius 3 around x is trivial. In this paper we show that these special situations cannot occur.

This paper presents some new results on algebraic K-theory with finite coefficients. The argument is based on a topological construction of a space $F_mK(R)$, for any ring $R$ and any integer $m\geq 2$, having the property that the ordinary homotopy theory of $F_mK(R)$ is isomorphic to the algebraic K-theory of $R$ with coefficients in $\bb {Z}/m$: $\pi_n(F_mK(R))\cong K_n(R;\bb {Z}/m)$ for $n\geq 1$. This space $F_mK(R)$ is called the $\mod m$ K-theory space of $R$. The paper is devoted to the investigation of several properties of the groups $K_n(R;\bb {Z}/m)$, for $n\in\bb {Z}$, and to some calculations of the integral homology of finite K-theory spaces.

We study a set $\mathcal{M}_{K,N}$ parameterising filtered SL(K)-Higgs bundles over $\mathbb{C}P^1$ with an irregular singularity at $z = \infty$, such that the eigenvalues of the Higgs field grow like $\vert \lambda \vert \sim \vert z^{N/K} \mathrm{d}z \vert$, where K and N are coprime. $\mathcal{M}_{K,N}$ carries a $\mathbb{C}^\times$-action analogous to the famous $\mathbb{C}^\times$-action introduced by Hitchin on the moduli spaces of Higgs bundles over compact curves. The construction of this $\mathbb{C}^\times$-action on $\mathcal{M}_{K,N}$ involves the rotation automorphism of the base $\mathbb{C}P^1$. We classify the fixed points of this $\mathbb{C}^\times$-action, and exhibit a curious 1-1 correspondence between these fixed points and certain representations of the vertex algebra $\mathcal{W}_K$ ; in particular we have the relation $\mu = {k-1-c_{\mathrm{eff}}}/{12}$ , where $\mu$ is a regulated version of the L2 norm of the Higgs field, and $c_{\mathrm{eff}}$ is the effective Virasoro central charge of the corresponding W-algebra representation. We also discuss a Białynicki–Birula-type decomposition of $\mathcal{M}_{K,N}$ , where the strata are labeled by isomorphism classes of the underlying filtered vector bundles.

We discuss a minimisation problem of the degree of the Chow–Mumford (CM) line bundle among all possible fillings of a polarised family with fixed general fibers, motivated by the study of the moduli space of K-stable Fano varieties. We show that such minimisation implies the slope semistability of the fiber if the central fiber is smooth.

Gross and Siebert developed a program for constructing in arbitrary dimension a mirror family to a log Calabi–Yau pair (X, D), consisting of a smooth projective variety X with a normal-crossing anti-canonical divisor D in X. In this paper, we provide an algorithm to practically compute explicit equations of the mirror family in the case when X is obtained as a blow-up of a toric variety along hypersurfaces in its toric boundary, and D is the strict transform of the toric boundary. The main ingredient is the heart of the canonical wall structure associated to such pairs (X, D), which is constructed purely combinatorially, following our previous work with Mark Gross. In the case when we blow up a single hypersurface we show that our results agree with previous results computed symplectically by Aroux–Abouzaid–Katzarkov. In the situation when the locus of blow-up is formed by more than a single hypersurface, due to infinitely many walls interacting, writing the equations becomes significantly more challenging. We provide the first examples of explicit equations for mirror families in such situations.

The forced torsional motion of a semi-infinite isotropic right-circular elastic cylinder is studied in this paper, the vibrations being excited by a harmonically oscillating rigid disc, securely attached to the plane face of the cylinder. Two distinctsituations are considered, namely, when the curved surface of the cylinder is clamped and when it is stress free. In both cases the problem is reduced to the solution of a Fredholm integral equation of the second kind which can be solved iteratively when the frequency of oscillation is small and the cylinder radius large. Using the iterative solutions, approximate expressions are deduced for the distribution of shear stress under the disc and the driving couple required. In an appendix a discussion is given of a mathematically similar problem in the theory of the swirling flow of a perfect fluid.

We show that certain graphs of groups with cyclic edge groups are aTmenable. In particular, this holds when each vertex group is either virtually special or acts properly and semisimply on ℍn.

Let N > 1 and let AN be the polydisc algebra, i.e. the algebra of all continuous functions on the closed polydisc δ¯N ⊂ N, analytic on the open polydisc δN, with sup norm. Call a closed set F ⊂ δ¯N a peak interpolation set for AN if given any f ε C(F), f ≠ 0, there is an extension f ε AN of f such that ¦f˜(z)¦ < ‖ f ‖ (z ε δ¯N - F); call F a norm preserving interpolation set for AN if given any f ε C(F) there is an extension f˜ ε AN of f such that ‖f˜‖ = ‖f‖. The paper gives a complete description of norm preserving interpolation sets for AN in terms of peak interpolation sets for AM, M ≤ N.

13. Formulae have been obtained for the moment of the fluid forces which act on a plane plate which is gliding on the surface of a stream of finite or infinite depth. It has been shown that the formulae for the lift and moment forces acting on a plate which is gliding on a stream of infinite depth may be obtained by a limiting process from the formulae for the general case of gliding on a stream of finite depth. Also, by another limiting process, the lift and moment forces for the Rayleigh flow past a flat plate in an infinite stream have been obtained from the lift and moment forces for gliding on a stream of infinite depth.

Numerical work has been carried out for the case when the angle of incidence of the plate to the stream is 5° and this has shown the variation of the lift and moment forces and the position of the centre of pressure with a range of values of the length of the plate, the depth of the stream and the height of the trailing edge of the plate above the bed of the stream.